I think, there is a very little difference, when using ganz or ziemlich. Ganz is a little bit stronger than ziemlich, but there isn't much space in between meaning. Most time the difference is vague and indecisive. I think most people will use them spontaneously equally without seeing any difference, though they know/feel a difference (I refer to the subconscious).
Mir geht es ganz schlecht
This sentence means, that I am feeling very sick. I am not able to do anything and I just want to stay in bed.
Mir geht es ziemlich schlecht
I am in a weak constitution, not healthy. I am feeling sick, but I am able to do the most necessary work (I can go to the baker, though I wish someone else did)
Du bist ein ganz starker Gegner
Your opponent is very strong and you need much endurance, concentration to beat him. You have to go a high pace, much deceptions, ...
Du bist ein ziemlich starker Gegner
Your opponent is very good but you are obviously better. It's not easy to beat him, you still have to fight, but you can allow yourself doing mistakes and forgive good chances without risking your win
Regarding your example (but I will turn to the opposite):
Es ist ganz (schön) starker Schneefall
First: The schön sounds better to me. I don't know if it just regional, but in this example I would never omit it. In this case it stress the expression, give it more meaning: very very much. But this just as a side note.
Now what I want to express saying this sentence. Let's imagine the situation that I am driving my car through a heavy snowfall. Then I mean: I don't see much. I've to drive very slowly and carefully. I have to concentrate very much on keeping the lane.
Es ist ziemlich starker Schneefall
Let's take the same situation. Now I give more expression on the fact, that it is disturbing, annoying. I still have to drive carefully, but I drive a lot faster, because it isn't that dangerous.
Both variation can also stress my astonishing how much snow falls.